Question

Find the area of the quadrant of the circle whose circumference is $44{\text{cm}}$.

Answer 1

Hint:
First, we will find the radius of the circle using the formula of the circumference of the circle. The formula of the circumference of the circle is given as-
$ \Rightarrow $ Circumference of the circle=$2\pi r$ where r is the radius of the circle
Now, we will find the area of the circle using the formula- the area of the circle=$\pi {r^2}$where r is the radius of the circle. Since quadrant is$\dfrac{1}{4}{\text{th}}$ of the circle so the area of quadrant will also be $\dfrac{1}{4}{\text{th}}$ of the area of the circle. So divide the obtained area by $4$ to get the answer.

Complete step by step solution:
Given, the circumference of the circle =$44{\text{cm}}$
We have to find the area of the quadrant of the circle.

Suppose the given circle has the circumference of $44{\text{cm}}$so its quadrant area will be ABC. From the diagram, it is clear that the quadrant of the circle is $\dfrac{1}{4}{\text{th}}$ of the circle so the area of quadrant will also be $\dfrac{1}{4}{\text{th}}$ of the area of the circle.
First, we will find the radius of the circle using the formula of the circumference of the circle which is given as-
$ \Rightarrow $ Circumference of the circle=$2\pi r$ where r is the radius of the circle
On putting the given values, we get-
$ \Rightarrow 44 = 2\pi r$
On rearranging, we get-
 $ \Rightarrow \dfrac{{44}}{{2\pi }} = r$
On solving, we get-
$ \Rightarrow \dfrac{{22}}{\pi } = r$
Now, we know that the area of the circle=$\pi {r^2}$
So on putting the value of r, we get-
$ \Rightarrow $ Area of the circle=$\pi \times \dfrac{{22}}{\pi } \times \dfrac{{22}}{\pi }$
On solving, we get-
$ \Rightarrow $ Area of the circle=$22 \times \dfrac{{22}}{\pi }$
On putting the value of pi constant, we get-
$ \Rightarrow $ Area of the circle=$22 \times \dfrac{{22}}{{22}} \times 7$
On solving, we get-
$ \Rightarrow $ Area of the circle=$22 \times 7{\text{c}}{{\text{m}}^2}$ -- (i)
Now, the area of the quadrant of the circle=$\dfrac{1}{4}{\text{th}}$ of the area of the circle
So, we can write-
$ \Rightarrow $Area of the quadrant of the circle=$\dfrac{{22 \times 7}}{4}{\text{c}}{{\text{m}}^2}$
On solving, we get-
$ \Rightarrow $Area of the quadrant of the circle=$\dfrac{{11 \times 7}}{2}{\text{ = }}\dfrac{{77}}{2}{\text{c}}{{\text{m}}^2}$
On further solving, we get-
$ \Rightarrow $Area of the quadrant of the circle=${\text{38}}{\text{.5c}}{{\text{m}}^2}$

Answer- The area of the quadrant of the circle= $38.5{\text{c}}{{\text{m}}^2}$

Note:
Here the student may get confused about how the quadrant of the circle is $\dfrac{1}{4}{\text{th}}$ of the circle. So it is very easy to calculate we know that the angle the quadrant area forms with the circle is a right angle and the angle of the circle is ${360^\circ }$ so to find the area just divide the angle of the quadrant from the angle of the circle-
$ \Rightarrow \dfrac{{90}}{{360}} = \dfrac{1}{4}$
Similarly, if we know the angle the part of the shaded area forms with the circle, we can easily find its area.